Star Wars had lightsabers before any anime did, but weapons similar to them are quite common in anime, so they’re the topic today.
First of all, they certainly can’t literally be light. Neither a normal beam of light nor a laser stops when it reaches a set length, or blocks other beams from crossing itself.
In Universal Century Gundam (and yes, I’m aware that the picture above is from a non-UC series), beam sabers are one of the main uses of the fictional Minovsky particle. (Not the Megaparticle used in every other weapon that has “beam” in the name.) Their actual capabilities are exactly like Star Wars lightsabers except ten times bigger. Being made of a particle with mass instead of light makes them somewhat less implausible, but the sword-shaped forcefield required is still probably more impressive than the weapon’s actual effect. And cutting through the handle of a beam saber should release all those particles in an explosion, but instead, they are treated as though the particles that make up the beam cease to exist if the saber stops functioning. (Which only makes sense if the half-life of a Minovsky particle is extremely short and a beam saber is having to constantly generate new ones…which would mean that it should require more power than the mobile suit itself.)
In Project Quintessence, I have plasma rods and plasma swords that are fairly similar in function, but their entire length is solid and they merely emit plasma out of small holes. I freely admit that they’re probably still the most unrealistic thing in the story not justified by aether, but I think they are more plausible than swords where the entire blade is plasma or “energy”.